Split Lines

My spouse, the professional philosopher, was sharing some of the engagingly wrong student responses. I hope it hasn’t shocked you to learn your instructors do this, but, if you got something wrong in an amusing way, and it was easy to find someone to commiserate with, yes, they said something.

The particular point this time was about Plato’s Analogy of the Divided Line, part of a Socratic dialogue that tries to classify the different kinds of knowledge. I’m not informed enough to describe fairly the point Plato was getting at, but the mathematics is plain enough. It starts with a line segment that gets divided into two unequal parts; each of the two parts is then divided into parts of the same proportion. Why this has to be I’m not sure (my understanding is it’s not clear exactly why Plato thought it important they be unequal parts), although it has got the interesting side effect of making exactly two of the four line segments of equal length.

Anyway, the student trying to explain this proposed a line divided into two equal halves, the only possible split that would go exactly against the text. Or as I put it, the student’s fraction — cutting the line in half, or to 0.5 if you like decimals more — was one she or he had an exactly zero percent chance of picking at random. That amused us both, even as we agreed it was unfair: after all, if you were asked to pick a number between zero and one, you’re probably going to pick a half, a third, a quarter, maybe two-thirds or three-quarters, and that’s about it. If you’re a mathematician maybe you get cute and pick \frac{1}{\sqrt{2}} or \frac{1}{e} , but, really, you’re picking not from the infinitely large set of possibilities but from a couple of easy-to-name numbers.

Also unfair: any one specific number has a probability of zero of being picked, if you’ve got an equal chance of picking all the numbers between the lower and upper bounds. There are several ways to argue this and I’d like to look at at least one of them.

First, let’s suppose that we’re picking a number between zero and one; let me call that x because it’s so much easier to have a letter for that. (We’re also making the assumption we can match the numbers between zero and one to the points on a line segment, but we do that with number lines so much that we barely notice we’re doing it anymore.) So the probability that this number turns out to be between zero and one has got to be 1; if we got something less than zero or greater than one then we did something terribly wrong.

Second, and I hope you’ll grant this, the probability that x turns out to be less than 1/2, should be 1/2. After all, half of all the numbers between 0 and 1 are less than 1/2. And the probability that x turns out to be greater than 1/2 should also be 1/2.

So, the chance that x is somewhere between 0 and 1 is 1. The chance that it’s less than 1/2 is 1/2, and the chance that it’s greater than 1/2 is also 1/2. So the chance that x is exactly 1/2 would then be 1 minus 1/2 minus 1/2, or, 0. And you can make pretty much the same argument to rule out every other number — an x of 1/3, or an x of \frac{1}{e} , or whatever you like — thus bringing out a second way our laughter was unfair.

You might not be convinced by this argument. I don’t blame you; it feels a little to me like there’s a card palmed in there somewhere, most likely in my talking about the probability that x is “less than” 1/2 instead of, oh, “less than or equal to” 1/2. We can go around the question in other ways that are maybe more convincing.